A normal distribution is a common assumption underlying many statistical tests. Although the assumption is usually not important, it is possible to test whether it is plausible. Specifically, the unit normal table lists proportions for individual sections of a normal distribution, and a chisquare test for goodness of fit can be used to evaluate whether a specific distribution fits the proportions. For example, if a normal distribution is divided into sections using zscore values, then proportions in each section should be as follows:
1.5 < z  0.5 <z  0.5 <z  
z < 1.5  0.5  0.5  1.5  z > 1.5 
6.68%  24.17%  38.30%  24.17%  6.68% 
Use these proportions to test whether the following sample of n = 90 scores is significantly different from a normal distribution. Test with a = .05.
1.5 < z  0.5 <z  0.5 <z  
z < 1.5  0.5  0.5  1.5  z > 1.5 
8  19  31  23  9 
To Test: If there is significant difference between the observed values and normal distribution.
Given info:
Observed values:





8  9  31  23  9 
Expected values:





6.68%  24.17%  38.30%  24.17%  6.68% 
Calculation:
The expected values can be rewritten as frequency as follows:
Thus, the expected values are as follows:
6.012, 21.753, 34.47, 21.753, 6.012
The following table summarizes the calculations:
Cell 


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